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A046969 Denominators of coefficients in Stirling's expansion for log(Gamma(z)).

%I A046969 #43 Feb 16 2025 08:32:39
%S A046969 12,360,1260,1680,1188,360360,156,122400,244188,125400,5796,1506960,
%T A046969 300,93960,2492028,505920,396,2418179400,444,21106800,3109932,118680,
%U A046969 25380,104700960,6468,324360,2283876,382800,40356,201025024200,732
%N A046969 Denominators of coefficients in Stirling's expansion for log(Gamma(z)).
%C A046969 From _Lorenzo Sauras Altuzarra_, Oct 13 2020: (Start)
%C A046969 Conjecture I: if n > 2, then a(A005382(n))/12 is prime.
%C A046969 Conjecture II: if a(n)/12 is prime, then a(n-1)/12 - (n-1), a(n)/12 - n and a(n+2)/12 - (n+2) are multiples of 6. (End)
%D A046969 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.
%D A046969 L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
%H A046969 Robert G. Wilson v, <a href="/A046969/b046969.txt">Table of n, a(n) for n = 1..1000</a>
%H A046969 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A046969 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.
%H A046969 Thomas Bayes, <a href="http://www.york.ac.uk/depts/maths/histstat/letter.pdf">A letter to John Canton</a>, Phil. Trans. Royal Society London, 53 (1763), 269-271.
%H A046969 R. P. Brent, <a href="http://arxiv.org/abs/1608.04834">Asymptotic approximation of central binomial coefficients with rigorous error bounds</a>, arXiv:1608.04834 [math.NA], 2016.
%H A046969 N. Elezovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Elezovic/elezovic5.html">Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers</a>, J. Int. Seq. 17 (2014) # 14.2.1.
%H A046969 C. Impens, <a href="http://www.jstor.org/stable/3647856">Stirling's series made easy</a>, Am. Math. Monthly, 110 (No. 8, 2003), pp. 730-735.
%H A046969 Gergő Nemes, <a href="https://doi.org/10.1080/10652469.2012.725168">Generalization of Binet's Gamma function formulas</a>, Integral Transforms and Special Functions, 24:8, pp. 597-606, 2013.
%H A046969 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingsSeries.html">Stirling's Series</a>
%F A046969 From denominator of Jk(z) = (-1)^(k-1)*Bk/(((2k)*(2k-1))*z^(2k-1)), so Gamma(z) = sqrt(2pi)*z^(z-0.5)*exp(-z)*exp(J(z)).
%p A046969 a := n -> denom(bernoulli(2*n)/(2*n*(2*n-1))): # _Lorenzo Sauras Altuzarra_, Oct 13 2020
%t A046969 Table[ Denominator[ BernoulliB[2n]/(2n(2n - 1))], {n, 31}] (* _Robert G. Wilson v_, Sep 21 2006 *)
%t A046969 s = LogGamma[z] + z - (z - 1/2) Log[z] - Log[2 Pi]/2 + O[z, Infinity]^62;
%t A046969 DeleteCases[CoefficientList[s, 1/z], 0] // Denominator (* _Jean-François Alcover_, Jun 13 2017 *)
%o A046969 (PARI) a(n)=if(n<1,0,denominator(bernfrac(2*n)/(2*n)/(2*n-1)))
%Y A046969 Numerators are given in A046968. Cf. A005382.
%K A046969 frac,nonn,nice
%O A046969 1,1
%A A046969 Douglas Stoll, dougstoll(AT)email.msn.com
%E A046969 More terms from _Frank Ellermann_, Jun 13 2001
%E A046969 Bayes reference from _Henry Bottomley_, Jun 03 2003